A magic square has been defined in the literature as an array of numerals arranged in the form of a square so that the sums of the numerals in each row, each column and the two main diagonals are equal in value known as the square's constant. The number of rows and columns must be greater than two. The literature suggests that this number is commonly three or four although up to nine rows and columns are discussed in books such as New Recreations With Magic Squares by Benson and Jacoby, 1976, Dover Publications, Library of Congress Catalog Card No. 74-28909 and Magic Square by Fults, 1974, The Open Court Publishing Company with reference No. 73-23041.
In many magic squares, the numerals are consecutive numbers each used once. When the consecutive numbers start with the integer 1, the square is called a pure, or traditional, magic square. The digits in a pure magic square of the third-order can be arranged in eight ways by rotations and reflections. But a magic square which does not start with the integer 1 allows for a larger number of different arrangements, and when non-consecutive numbers are allowed, a massive number of possible arrangements exist. The literature suggests that for fourth-order magic squares exactly 880 magic square arrangements exist and that over 549,000 different orders have been found for a fifth-order magic square, although several million are believed possible.